Sur les corps de Hilbert-Speiser

Herreng, Thomas

doi:10.5802/jtnb.519

Cited by 7 publications

(7 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We can often verify part (b) of Proposition 1.8 by using the results of [1] and [2]. In other cases, we can quote existing results in the literature showing that K is not Hilbert-Speiser of type C p (see [12], [14]). Combining all of these results gives the following theorem.…”

Section: We Observe Immediately That Trmentioning

confidence: 84%

See 1 more Smart Citation

On the restricted Hilbert–Speiser and Leopoldt properties

Byott¹,

Carter²,

Greither³

et al. 2011

Illinois J. Math.

View full text Add to dashboard Cite

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if, for every tame G-Galois extension L/K, the ring of integers O L is free as an O K [G]-module. If O L is free over the associated order A L/K for every G-Galois extension L/K, then K is called a Leopoldt field of type G. It is well-known (and easy to see) that if K is Leopoldt of type G, then K is Hilbert-Speiser of type G. We show that the converse does not hold in general, but that a modified version does hold for many number fields K (in particular, for K/Q Galois) when G = C p has prime order. We give examples with G = C p to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.

show abstract

Section: We Observe Immediately That Trmentioning

confidence: 84%

“…To prove the claim, we apply Herreng's formula [14,Proposition 3.2] for the 3-rank d 3 ((O K /3O K ) × ) of the unit group of the residue ring O K /3O K . Writing f p for the inertia degree of p, this yields…”

Section: Realisable Classes and The Proof Of Proposition 18mentioning

confidence: 99%

On the restricted Hilbert–Speiser and Leopoldt properties

Byott¹,

Carter²,

Greither³

et al. 2011

Illinois J. Math.

View full text Add to dashboard Cite

show abstract

“…We continue the investigation of this case by establishing the following result, the proof of which is based on a detailed analysis of locally free class groups and ramification indices. Theorem 1.1 can be seen as an analogue of the following result of Herreng (see [Her05,§3]). The authors are grateful to Nigel P. Byott for pointing out that the original hypothesis that K/Q is Galois can be weakened as below.…”

Section: Introductionmentioning

confidence: 92%

“…By fixing a finite abelian group G one can consider a finer problem: given a number field K, does every tame G-Galois extension L/K have a normal integral basis? If so, K is said to be a Hilbert-Speiser field of type G. The simplest case to consider is when G = C p , the cyclic group of prime order p. This has been studied, for instance, in [Car03], [Car04], [Her05], [Ich02], [Ich04], [Ich07a], [Ich07b] and [IST07]. We continue the investigation of this case by establishing the following result, the proof of which is based on a detailed analysis of locally free class groups and ramification indices.…”

Section: Introductionmentioning

confidence: 99%

On totally real Hilbert–Speiser fields of type C_p

Greither

Johnston²

2009

Acta Arith.

View full text Add to dashboard Cite

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers O L is free as an O K G-module. Let C p denote the cyclic group of prime order p. We show that if p ≥ 7 (or p = 5 and extra conditions are met) and K is totally real with K/Q ramified at p, then K is not Hilbert-Speiser of type C p .

show abstract

“…To sum up: in the context of C l -Leopoldt fields K that intersect nontrivially with Q(ζ l ), if we just use ad hoc arguments and previous results of Ichimura and Sumida-Takahashi it appears that we are able to obtain restrictions on the class number of Q(ζ l ) only in the case when [Q(ζ l ) : K ∩ Q(ζ l )] is equal to 2 or an odd number, and thus finiteness conditions if we assume the generalized Riemann hypothesis, by [AD03] (without fixing l if [Q(ζ l ) : K ∩ Q(ζ l )] is 1, 2 or 3). Returning to Hilbert-Speiser fields, from [Her05] we already knew that we do not have such odd cases of C l -Hilbert-Speiser fields for l ≥ 5, unless the intersection is Q( √ −l) when l ≡ 3 (mod 4). When instead [Q(ζ l ) : K ∩ Q(ζ l )] is even, i.e.…”

Section: Complete List Of Real Abelianmentioning

confidence: 99%

Tame Galois module structure revisited

Ferri

Greither

2019

Annali di Matematica

View full text Add to dashboard Cite

In the last decades a lot of work has been done in the study of Hilbert-Speiser fields; in particular, criteria were developed which insure that a given field is not Hilbert-Speiser. The framework is Galois module structure of extensions of number fields. More precisely the object of our study is the property of an extension of number fields to have a normal integral basis: an extension L/K of number fields admits a NIB if O L is a rank 1 free O K [G]-module. The Hilbert-Speiser theorem states that for abelian number fields tameness is equivalent to the existence of a normal integral basis over Q, while in general we only have that an extension with NIB has to be tame. We will say that a number field K is Hilbert-Speiser if every tame abelian extension L/K has NIB, and C l -Hilbert-Speiser if every such extension with Galois group isomorphic to C l , the cyclic group of prime order l, has NIB.The first contribution to the topic of Hilbert-Speiser fields came from the important result contained in [GRRS99]: Q is the only Hilbert-Speiser field, i.e. for every number field K Q there exists a (cyclic of prime order) tame abelian extension that does not have NIB. Subsequent research went towards the finer problem of finding crieria for C l -Hilbert-Speiser fields. For instance, in [Car03], [Ich04] and [Yos09] there are some conditions for abelian C 2 and C 3 -Hilbert-Speiser fields, while from [Her05] and [GJ09] we know that in general C l -Hilbert-Speiser fields cannot be highly ramified if they are respectively totally imaginary or totally real.In this work, our intention is to consider a weakened version of NIB: we will say that a tame abelian extension L/K of number fields has a weak normal integral basis if M ⊗ OK[G] O L is free of rank 1 over M, where M is the maximal order of K [G]. WNIB's have been studied for instance in [Gre90], [Gre97] and [GJ12]. We shall ask the same questions as above substituting "WNIB" to "NIB" everywhere, and the condition that results from this substitution will be called "Leopoldt field" instead of "Hilbert-Speiser field". We are going to find mainly necessary conditions for number fields to be C l -Leopoldt, where as before l is a prime number, which also give criteria and (sometimes conditional) finiteness results for C l -Hilbert-Speiser fields; for instance we will see that this permits to correct an oversight contained in the article [Ich16], whose techniques, even though they were originally conceived to deal with Hilbert-Speiser fields, turn out to be supple enough to be applied to our problem as well.

show abstract

Sur les corps de Hilbert-Speiser

Cited by 7 publications

References 4 publications

On the restricted Hilbert–Speiser and Leopoldt properties

On the restricted Hilbert–Speiser and Leopoldt properties

On totally real Hilbert–Speiser fields of type C_p

Tame Galois module structure revisited

Contact Info

Product

Resources

About

Sur les corps de Hilbert-Speiser

Cited by 7 publications

References 4 publications

On the restricted Hilbert–Speiser and Leopoldt properties

On the restricted Hilbert–Speiser and Leopoldt properties

On totally real Hilbert–Speiser fields of type Cp

Tame Galois module structure revisited

Contact Info

Product

Resources

About

On totally real Hilbert–Speiser fields of type C_p